Stiffness and Accuracy in the Method of Lines Integration of Partial Differential Equations. Part I: Introductory Discussion.
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The Method of Lines is discussed in terms of a general, linear, constant coefficient partial differential equation which is first and second order in both a temporal initial value independent variable and a spatial boundary value independent variable. Four special cases of the general partial differential equation are considered. For each of the four problems, the approximating system of first order, initial value ordinary differential equations was studied numerically in terms of a conventional eigenvalue analysis of the associated Jacobian matrix. A comparison of the eigenvalues for the approximating ordinary differential equations with the exact eigenvalues of the partial differential equations gives a direct indication of the accuracy of the finite difference approximation of the spatial derivatives. Two conventional finite difference approximations are compared. The stiffness i.e., stability characteristics of the approximating ordinary differential equations are also considered in terms of the ratio of the largest to smallest eigenvalues of the Jacobian matrix. Modified author abstract
- Theoretical Mathematics